• Journal of the Korean School Mathematics Society
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• v.25 no.2
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• pp.105-124
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• 2022

The contents of fraction division in textbooks are important because there were changes in situations and problem solving methods in textbooks according to the revision of the curriculum and the contents of textbooks affect students' learning directly. So, this study analyzed the achievement standards of the curriculum and formula types and situations, and the introduction process of non-standard and standard algorithms presented in Korean mathematics textbooks. The results are follows: there was little difference in the achievement standards of the curriculum, but there was a difference in the arrangement of contents by grades in textbooks. There was a difference in the types of formula according to textbooks. And the situation became more diverse; recent textbooks have changed to the direction of using the non-standard and the standard algorithm in parallel. In conclusion, I proposed categorizing rather than splitting the types of fraction division, the connection of non-standard and standard algorithm, and the need to prepare methods to pursue generalization and justification according to the common characteristics in the process of introducing standard algorithm.

• Journal of Elementary Mathematics Education in Korea
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• v.18 no.3
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• pp.425-439
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• 2014

The concept of a fraction as division is a core idea which serves as a axiom in the process of a extension of the natural number system to rational number system. Also, it has necessary position in elementary mathematics. Nevertheless, the timing and method of the introduction of this concept in Korean elementary math textbooks is not well established. In this thesis, I suggested a solution of a various topics which is related to this problem, that is, transforming improper fraction to mixed number, the usage of quotient as a term, explaining the algorithm of division of fraction, transforming fraction to decimal.

• Journal of Educational Research in Mathematics
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• v.23 no.1
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• pp.17-35
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• 2013

The purpose of this study was to analyze the extendibility of division algorithm and Euclid algorithm for integers to algorithms for rational numbers based on word problems of fraction division. This study serviced to upgrade professional development of elementary and secondary mathematics teachers. In this paper, fractions were used as expressions of rational numbers, and they also represent rational numbers. According to discrete context and continuous context, and measurement division and partition division etc, divisibility was classified into two types; one is an abstract algebraic point of view and the other is a generalizing view which preserves division algorithms for integers. In the second view, we raised some contextual problems that can be used in school mathematics and then we discussed division algorithm, the greatest common divisor and the least common multiple, and Euclid algorithm for fractions.

• School Mathematics
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• v.11 no.1
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• pp.71-91
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• 2009

The purpose of this study was to analyze operation sense in detail with regard to division of fraction. For this purpose, two sixth grade students who were good at calculation were clinically interviewed three times. The analysis was focused on (a) how the students would understand the multiple meanings and models of division of fraction, (b) how they would recognize the meaning of algorithm related to division of fraction, and (c) how they would employ the meanings and properties of operation in order to translate them into different modes of representation as well as to develop their own strategies. This paper includes several episodes which reveal students' qualitative difference in terms of various dimensions of operation sense. The need to develop operation sense is suggested specifically for upper grades of elementary school.

• Journal of Elementary Mathematics Education in Korea
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• v.17 no.3
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• pp.431-455
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• 2013

In this thesis, I classified various meanings of fraction into two categories, i.e concept(rate, operator, division) and model(whole-part, measurement, allotment), and surveyed appearances which is shown in Korea elementary mathematics textbook. Based on this results, I derived several implications on learning-teaching of fraction in elementary education. Firstly, we have to pursuit a unified formation of fraction concept through a complementary advantage of various concepts and models Secondly, by clarifying the time which concepts and models of fraction are imported, we have to overcome a ambiguity or tacit usage of that. Thirdly, the present Korea's textbook need to be improved in usage of measurement model. It must be defined more explicitly and must be used in explanation of multiplication and division algorithm of fraction.

• Computational Structural Engineering : An International Journal
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• v.1 no.1
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• pp.21-29
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• 2001

The general topology optimization can be considered as optimal material distribution. Such an approach can be unstable, unless composite materials are introduced. In this research, a nodal volume fraction method is used to obtain the optimum topology of continuum structures. This method is conducted from a composite material model composed of isotropic matter and spherical void. Because the appearance of the chessboard patterns makes the interpretation of the optimal material layout very difficult, this method contains a chessboard prevention strategy. In this research, several topology optimization problems are presented to demonstrate the validity of the present method and the recursive quadratic programming algorithm is used to solve the topology optimization problems.

• Journal of Elementary Mathematics Education in Korea
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• v.20 no.3
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• pp.371-391
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• 2016

Long division was one of the major issues in math war II started from 1990's in US. In this paper, we investigated this concretely and examined present teaching condition of long division in Korea. Firstly, Long division is not only a mechanical way to get the answer but also a realization of core conception in elementary mathematics. Futhermore it is a connecting link between elementary and middle mathematics education. Secondly, it is needed to use the term 'long division' to provide the concrete teaching guidelines. Thirdly, a minor algorithm, like an 'partial quotient method', is necessary to introduce in order to help understanding long division.

• School Mathematics
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• v.7 no.2
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• pp.103-121
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• 2005

This article compares and analyzes mathematics textbooks of North Korea, South Korea, China and Japan and draws meaningful ways for introducing the division algorithm of fractions. The analysis is based on the five contexts: 'measurement division', 'determination of a unit rate', 'reduction of the quantities in the same measure', 'division as the inverse of multiplication or Cartesian product', 'analogy with multiplication algorithm of fractions'. The main focus of the analysis is what context is used to introduce the algorithm and how much it can appeal to students. This analysis supports that there is a few differences of introducing methods the division algorithm of fractions among those countries and more meaningful way can be considered than ours. It finally suggests that we teach the algorithm in a way which can have students easily see the reason of multiplying the reciprocal of a divisor when they divide with fractions. For this, we need to teach the meaning of a reciprocal of fraction and consider to use the context of determination of a unit rate.

• Education of Primary School Mathematics
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• v.27 no.1
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• pp.91-110
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• 2024

The purpose of this study is to explore visual models for deriving the fractional division algorithm, to see how students understand this integrated model, the rectangular partition model, when taught in elementary school classrooms, and how they structure relationships between fractional division situations. The conclusions obtained through this study are as follows. First, in order to remind the reason for multiplying the reciprocal of the divisor or the meaning of the reciprocal, it is necessary to explain the calculation process by interpreting the fraction division formula as the context of a measurement division or the context of the determination of a unit rate. Second, the rectangular partition model can complement the detour or inappropriate parts that appear in the existing model when interpreting the fraction division formula as the context of a measurement division, and can be said to be an appropriate model for deriving the standard algorithm from the problem of the context of the inverse of a Cartesian product. Third, in the context the inverse of a Cartesian product, the rectangular partition model can naturally reveal the calculation process in the context of a measurement division and the context of the determination of a unit rate, and can show why one division formula can have two interpretations, so it can be used as an integrated model.

• Journal of Communications and Networks
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• v.15 no.1
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• pp.102-110
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• 2013

This paper considers a cooperative orthogonal frequency division multiple access (OFDMA)-based cognitive radio network where the primary system leases some of its subchannels to the secondary system for a fraction of time in exchange for the secondary users (SUs) assisting the transmission of primary users (PUs) as relays. Our aim is to determine the cooperation strategies among the primary and secondary systems so as to maximize the sum-rate of SUs while maintaining quality-of-service (QoS) requirements of PUs. We formulate a joint optimization problem of PU transmission mode selection, SU (or relay) selection, subcarrier assignment, power control, and time allocation. By applying dual method, this mixed integer programming problem is decomposed into parallel per-subcarrier subproblems, with each determining the cooperation strategy between one PU and one SU. We show that, on each leased subcarrier, the optimal strategy is to let a SU exclusively act as a relay or transmit for itself. This result is fundamentally different from the conventional spectrum leasing in single-channel systems where a SU must transmit a fraction of time for itself if it helps the PU's transmission. We then propose a subgradient-based algorithm to find the asymptotically optimal solution to the primal problem in polynomial time. Simulation results demonstrate that the proposed algorithm can significantly enhance the network performance.